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# The diagonal of a rectangle is thrice its smaller side. Find the ratio of its sides.A) $\sqrt 2 :1$B) $2\sqrt 2 :1$C) $3:2$D) $\sqrt 3 :1$

Last updated date: 20th Jun 2024
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Hint: A ratio says how much of one thing there is compared to another thing.
A rectangle has two diagonals. Each one is a line segment drawn between the opposite vertices (corners) of the rectangle. The diagonals have the following properties:
The two diagonals are congruent (same length).
Each diagonal bisects the other. In other words, the point where the diagonals intersect (cross), divides each diagonal into two equal parts.
Each diagonal divides the rectangle into two congruent right triangles. Because the triangles are congruent, they have the same area, and each triangle has half the area of the rectangle.

Let us assume that the length of the smaller side of the rectangle,
i.e., BC be x and
length of the larger side , i.e., AB be y.

It is given that the length of the diagonal is three times that of the smaller side,
the diagonal is 3x
Now, applying Pythagoras theorem, we get:
(Diagonal)2 = (Smaller side)2 + (Larger side)2
The diagonal of the rectangle is the hypotenuse of its sides.
$\Rightarrow {x^2} + {y^2} = {(3x)^2} \\ \Rightarrow {x^2} + {y^2} = 9{x^2} \\ \Rightarrow {y^2} = 8{x^2} \\ \Rightarrow y = 2\sqrt 2 x \\$
So, the longer side is $2\sqrt 2 x$
The ratio of sides larger: smaller = $y:x = 2\sqrt 2 :1$ ​
So, option (B) is correct answer

Note: So, if you are given two values you can always find the third in case of length, breadth and diagonal for rectangle. Similarly given the two values of length, breadth and diagonal for rectangle then you can always find area and perimeter for rectangle.